Article Review…
A Critical Question: Why Can’t Mathematics Education
and History of Mathematics Coexist?
Kevin Nooney
Fried, M. N. (2001). Can mathematics education and history of mathematics coexist? Science and Education,
10(4), 391-408.
claims that mathematics educators are so
Fried’s abstract of his article (p. 391):
committed to modern mathematics and modern
Despite the wide interest in combining mathematics
mathematical methods that any attempt to either
education and the history of mathematics, there are
incorporate
grave and fundamental problems in this effort. The
main difficulty is that while one wants to see historical
topics in the classroom or an historical approach in
teaching, the commitment to teach modern
mathematics and modern mathematics techniques
necessary in pure and applied sciences forces one
either to trivialize history or to distort it. In particular,
this commitment forces one to adopt a “Whiggish”
approach to the history of mathematics. Two possible
resolutions of the difficulty are (1) “radical separation”
– putting the history of mathematics on a separate track
from the ordinary course of instruction, and (2)
“radical accommodation” – turning the study of
mathematics into the study of mathematical texts.
Michael Fried makes a confusing case that
combining history of mathematics with
mathematics education is inherently difficult, if
not impossible. Initially Fried’s argument seems to
rest on the argument that mathematics educators’
commitment to modern mathematics makes
mathematics education incompatible with the
history of mathematics. However, the bulk of
Fried’s discussion concerns the purposes of the
historian: Fried asserts that the concerns of the
historian render education incompatible with the
history of mathematics. In this review I attempt to
expose Fried’s unsupported assertions by posing
questions that need to be addressed in order to fill
out his argument. Specifically, I question his
assumption that mathematics educators are
unavoidably committed to so-called modern
mathematics and his claim that the historian is
committed to limit his research to understanding
idiosyncrasies in the thinking of historical
mathematicians.
Fried states that there is no room for the history of
mathematics in mathematics education; he also
Vol. 12 No. 1, Spring 2002
historical topics or take an historical approach to
mathematics teaching is bound to compromise the
“true” history of mathematics. For him, most
discussions that support an historical approach to
teaching mathematics are guilty of endorsing bad
history, or more correctly a “Whiggish”
history—an anachronistic reading of history in
which modern mathematical concepts are ascribed
to ancient thinkers.i But, Fried asserts, even if this
problem of bogus history could be overcome, the
commitment to modern mathematics, mathematics
which have been proven best for solving modern
problems, leaves no room for examining what true
history of mathematics must focus on, namely the
idiosyncratic thinking of historical figures. This
idiosyncratic thinking not only established
mathematics completely different from our own,
but often led to many “dead ends” and numerous
mistakes by historical mathematicians. The
historian of mathematics wants and is duty bound
to examine the differences between historical
mathematics and modern mathematics. The
mathematics educator wants and is duty bound to
explain and justify modern methods. The purposes
Fried assumes for the historian and the
mathematics educator are at such odds as to render
any reconciliation impossible, or gravely difficult.
Fried takes a such a strong stance against
reconciliation throughout his argument that one
has to wonder what is left for him to support when
he closes by saying that his critical examination of
“attempts to introduce the history of mathematics
in mathematics education should not be
interpreted as opposing such attempts” (p. 406, his
emphasis).
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I find Fried’s argument confusing and often
contradictory. His discussion is deceptively simple
and it is in attempting to analyze his position that
confusions arise. For example, Fried proposes two
solutions to linking mathematics education to the
history of mathematics which are based on current
attempts that he assumes early in his discussion to
be doomed to failure. The apparent contradictions
of his argument lie in the many unsupported
claims he makes about the nature of mathematics
education and the history of mathematics. In this
review, rather than argue against his position, I try
to expose these unsupported claims and raise
questions that I feel Fried should answer before I,
or any one else, consider his position well
developed and worth accepting.
Fried takes the failure to implement historical
approaches in school mathematics as a signal to
consider whether it is possible to combine history
of mathematics and mathematics education; in this
paper he does not consider the possibility of
external sources of failure to implement those
approaches. Such sources might include the sociopolitical contexts of mathematics curricula
development, the depth of familiarity of
mathematics educators with historical topics, the
availability of historical materials for educators,
etc. I agree that Fried’s concern about the
possibility of combining history of mathematics
with mathematics education is important.
However, he takes the position that there is an
inherent difficulty in attempting such a
combination. I question this position because it
rests on so many unsupported claims.
Fried begins by classifying the recommendations
made by advocates of incorporating history into
mathematics curricula into two basic strategies. As
a result of the educational commitment that Fried
claims, both of the two basic strategies for
introducing history into school mathematics
programs are bound to fail. The first strategy, what
he calls the strategy of addition, involves
“historical anecdotes, short biographies, isolated
problems, and… does not alter a curriculum
except by enlarging it” (p. 392). This strategy is
bound to fail because mathematics programs are
already over-crowded and allow little or no room
for historical enhancement. The strategy of
accommodation is more thoroughgoing and
demands a restructuring of the mathematics
Vol. 12 No. 1, Spring 2002
curriculum based on historical development and
circumstances. This strategy is bound to fail
because it forces either an anachronistic reading of
history or a history so deeply edited as to be
similarly bastardized.
Fried assumes and asserts his position regarding
the commitments of the modern mathematics
educator without making a case for accepting that
assumption: Mathematics educators have an
“unavoidable commitment to the teaching of
modern mathematics and modern mathematical
techniques” (p. 392, his emphasis). I believe Fried
needs to address at least two major questions:
Why should we accept that mathematics educators
are committed exclusively to modern
mathematics? Why should we accept that such a
commitment is unavoidable? Fried might respond
that the answers to these two questions are
obvious from the purpose of mathematics
education; however what Fried suggests as the
sole purpose of mathematics education is
questionable. I will return to this point shortly.
In his closing comments, Fried quotes and accepts
Thomas Tymocsko’s claim that “pure
mathematics is ultimately humanistic
mathematics, one of the humanities, because it is
an intellectual discipline with a human perspective
and a history that matters” (p. 406). ii Fried
compares mathematics to literature, art, and
music; they are all expressions of “that vision and
inventiveness so much part [sic] of the human
spirit” (p. 406). He then states that the “study of
the history of mathematics is an effort to grasp this
facet of human creativity” (p. 406). His argument
suggests that it is only through the history of
mathematics that one can attempt to grasp
mathematics as a creative endeavor. Would he
also suggest that one can grasp art, literature, and
music as creative endeavors only through studying
their histories? I wonder what Fried conceives of
as suitable education in art, literature, or music;
would he demand a strong separation of studio arts
and history of art in a way parallel to the cleft he
sees between mathematics education and history
of mathematics? Would Fried consider courses in
studio arts to constitute art education and not those
in art history? Would he claim that there is no
room for art history in art education? While I can
imagine that discussions and debates are waged
over the roles and relative merit of studio arts and
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history of art courses, I suspect that most of us,
and most arts educators, conceive of studio arts
and art history as kinds of art education not as
kinds of education. Fried seems to assume that
learning mathematics and learning about history of
mathematics require two separate kinds of
education.
Questions arise, then, about Fried’s conceptions of
mathematics education and mathematics history.
He is much clearer about the purpose of the
history of mathematics than he is of the purpose of
mathematics education. We have to infer both
from his claims about the aims and commitments
of the mathematics historian versus those of the
mathematics educator. The historian is committed
to “understand the thought of the past” (p. 398), to
understand and examine the “idiosyncrasy of a
mathematician’s thought or of the thought of the
mathematician’s time” (p. 400). The mathematics
educator, he suggests, is committed only to
preparing future scientists and engineers.
Ultimately, in Fried’s view, these differing
commitments prevent an amicable marriage of
history and education in mathematics.
The historian’s commitment forces the
“serious” historian to always begin with an
assumption that the past is different than the
present and to focus on this difference. The
purpose, then, of the historian of mathematics is to
study peculiarities of mathematics—what, for
instance, makes a text “peculiarly Apollonian or
peculiarly Greek” (p. 400, his emphasis). This
study of peculiarities is what makes the historian
particularly interested in the “dead ends”
mathematicians come to and the mistakes they
make, for these are the kinds of things that
reveal the peculiarity of the person’s thought;
these are the things that reveal the human
character of doing mathematics. So, while one
might succeed in making mathematics
interesting, understandable, and approachable,
or in providing insights into concepts, problems
and problem-solving without history or with an
“unhistorical” history, humanizing mathematics
with history requires that history be taken quite
seriously, not as a mere tool, but as something
studied earnestly (pp. 400-401, his emphasis).
There are at least three issues at stake in Fried’s
claim. One is the foundational assumption Fried
takes all historical work to rest upon—the
Vol. 12 No. 1, Spring 2002
assumption that the past is unlike the present.
Another is the focus of study Fried seems to
demand that all historians take—the study of
idiosyncratic thinking peculiar to a particular
historical figure in a particular historical time
frame. Also at stake is the status of historical work
in education—that proper history is not to be
relegated to being a mere pedagogical tool.
If all historical work rests upon the assumption
that the past is different than the present, and rests
only upon that assumption as Fried suggests, of
what value can history have? If we assume that the
experiences of people in bygone times are
completely different than our own, what could we
hope to gain by examining their experiences?
Couldn’t anyone claim that dealing with our own
present day experience is difficult enough without
shouldering the burden of trying to understand the
disconnected and unrelated experiences of
someone in another era? And if their experiences,
their times, and their way of thinking and
understanding is completely disconnected and
dissimilar from ours, how can anyone from our
time claim to understand them in theirs? Fried
draws an analogy between mathematics education
and teaching literature by saying that “while one
learns something about Elizabethan culture by
reading Shakespeare, the main reason one reads
Shakespeare’s works is that they are great in their
own right” (p. 401). If there is nothing to be found
in common between the Merchant of Venice and
the shopkeeper on Main Street, what is the basis
for claiming that Shakespeare’s works are great in
their own right? While jokes about codpieces may
be lost on the modern reader or viewer, certainly
those same modern readers can understand
Hamlet’s frustration and sense of being betrayed
by those around him. Without some sense of
relevance to the modern viewer, Hamlet would
simply be a collection of odd movements and
sayings.
With Fried’s insistence that the historian’s primary
concern is with time dependent peculiarities, what
prevents history from becoming little more than a
collection of exotic trinkets? (Perhaps though, that
is all history is for Fried.) If the only criteria for,
or if the fundamental assumption of, the historian
is difference, what counts as different? What, and
where, is the line that demarcates the past and the
present? Do we rely on arbitrary boundaries in
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terms of years—those who lived within, say, 200
years of the present are assumed to be sufficiently
like us to be considered us? Or is there some
objective measure or process by which the
historian can establish the difference between
them and us? It seems that for history to have any
relevance, we must assume that the past is, after
all, somewhat like the present: there must be
something from the past that is translatable to the
present. While I sympathize with Fried’s concern
about anachronistic interpretations, I suggest that
any historical work that ignores either the
similarities or dissimilarities between the past and
present and focuses exclusively on one or the
other will be severely limited and dramatically
incomplete.
Fried claims that any history in which the present
is a measure of the past is bad history, or worse,
hardly even history at all. My question is how can
the present not be a measure of the past? Fried
rejects the search for the origins of ideas as a
major and fundamentally misconceived task for
history; in seeking the origins of concepts used in
modern mathematics, we will inevitably take away
the thoughts of the historical mathematician and
“make him think our own” (p. 396); which is to
say that we will read into ancient texts modern
concepts that were inconceivable at the time. For
Fried, it is the Whig history that traces paths (or a
single path) from the past to the present, while the
truly historical perspective “is the zigzag path of a
wanderer who does not know exactly where he is
going” (p. 396). But Fried’s opposition to reading
direction in history seems to blind him to the fact
that we cannot but examine the past from our own
position in the present. The very things that Fried
claims are of most interest to the historian of
mathematics—“the ‘dead ends’ mathematicians
come to and the mistakes they made” (p.
401)—can only be determined as “dead ends” and
“mistakes” from the stance of present day
mathematical theory and practice. Similarly, what
Fried calls modern methods and approaches (that
the mathematics educator is unavoidably
committed to) and justifies as “the most powerful
means to solve problems of interest and of
importance in the modern world” (p. 405, his
emphasis) can only be judged “the most powerful”
within a historical context. Whatever non-modern
methods might be, they proved less fruitful only
Vol. 12 No. 1, Spring 2002
for the kinds of problems Fried assumes
students will eventually face. How can he be sure that
they might not be fruitful in the future? Shouldn’t
students be aware, then, of the mathematics that did not
survive in order to enhance their appreciation of the
mathematics that educators demand they know?
Fried seems to agree that perhaps they should, but
he sees only two possibilities, both of which are
extensions of the strategies he previously
determined are doomed to failure. One solution is
a radical accommodation—students would learn
mathematics by engaging directly with historical
texts (as is done in certain “great books”
programs.) The other is a radical addition or
radical separation—students would have a history
of mathematics track parallel to their standard
mathematics courses. These “radical” solutions
seem to have their basis in Fried’s aversion to the
“use” of history in mathematics education. Fried is
very concerned that history is to be studied in its
own right—he seems to deny the same for
mathematics.
Fried’s conception of mathematics education
appears to be limited to delivering the useful,
powerful mathematics that will prepare competent
scientists and engineers. In fact Fried most
strongly implies this limited view of mathematics
education when he questions whether the
humanizing of mathematics through the radical
accommodation approach “satisfies the other
component of the mathematics teacher’s
commitment, namely, that students learn to do the
mathematics of science and engineering” (p. 401).
Does Fried really expect us to accept that the sole
purpose of mathematics education is to prepare
potential scientific workers in the applications of
mathematics? Are we to accept both that
mathematics is a human endeavor, with a history
that matters, and that mathematics is only for the
use of scientists and engineers? Would Fried
expect a mathematician not to be offended by the
implication of his claim that history is not to be
used but that mathematics is? Fried claims
(probably rightly so) that the commitments of the
mathematics educator and of the historian of
mathematics make the relationship of each to the
history of mathematics quite different (p. 398).
Can we not make the same claim about the
differences of commitments (and hence the
differences in relationship to mathematics)
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between the mathematician and the mathematics
educator? If we take Fried’s suggestion that we
radically separate mathematics education from the
history of mathematics (and hence separate the
mathematics educator from the historian), should
we demand a similar separation between the
mathematics educator and the practicing
mathematician?
Ultimately, Fried’s argument seems to more about
territorial boundaries than about the possibility of
infusing mathematics education with historical
understanding. Fried examined the mathematics of
Apollonius in his doctoral dissertation, and when
all is said and done in his case against history in
mathematics education, he appears to be a
historian trying to preserve the sanctity of his
esoteric work from being directed toward any kind
of utilitarian purpose. Fried’s article abstract and
Vol. 12 No. 1, Spring 2002
opening remarks lead the reader to expect his case
to be that the foundational commitments of the
mathematics educator render the history of
mathematics incompatible with mathematics
education. As I read his argument, struggling to
make sense of what appear to be his contradictory
leanings, I suspect that the case he actually wants
to make is that the commitments of the historian
render mathematics education incompatible with
the history of mathematics.
1
Fried adopts the term “Whig history” from Butterfield, H.
(1931/1951). The Whig Interpretation of History. New York:
Charles Scribner’s Sons.
1
Thomas Tymocsko was a philosopher who advocated a quasiempiricist and fallibilist view of mathematics.
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i
Fried adopts the term “Whig history” from Butterfield, H. (1931/1951). The Whig Interpretation of History. New
York: Charles Scribner’s Sons.
ii
Thomas Tymocsko was a philosopher who advocated a quasi-empiricist and fallibilist view of mathematics.
Vol. 12 No. 1, Spring 2002
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